Alas no. PSADT (or any other solution) does not magically give the local user Admin rights to install an application.
While running as SYSTEM, you can install all the bits requiring Admin permissions to install and then it can launch something running as the user to install the bits that only the user can do. There are different methods: Installing in user profiles - ActiveSetup vs. Execute-ProcessAsUser - The Toolkit / Tips & Tricks - PSAppDeployToolkit Community
For the “post-installation reboot/defer prompts” PSADT will detect if there is a user or not and display accordingly.